We explore a stochastic model that enables capturing external influences in two specific ways. The model allows for the expression of uncertainty in the parametrisation of the stochastic dynamics and incorporates patterns to account for different behaviours across various times or regimes. To establish our framework, we initially construct a model with random parameters, where the switching between regimes can be dictated either by random variables or deterministically. Such a model is highly interpretable. We further ensure mathematical consistency by demonstrating that the framework can be elegantly expressed through models taking the form of standard jump diffusions. Additionally, we consider a Markov-modulated approach for the switching between regimes characterised by random parameters. For all considered models, we derive characteristic functions, providing a versatile tool with wide-ranging applications. In a numerical experiment, we apply the framework to the financial problem of option pricing. The impact of parameter uncertainty is analysed in a two-regime model, where the asset process switches between periods of high and low volatility imbued with high and low uncertainty, respectively.

中文翻译：

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具有随机系数和制度之间切换的一致资产建模

我们探索了一种随机模型，该模型能够以两种特定方式捕获外部影响。该模型允许表达随机动力学参数化中的不确定性，并结合模式来解释不同时间或制度下的不同行为。为了建立我们的框架，我们首先构建一个具有随机参数的模型，其中状态之间的切换可以由随机变量或确定性决定。这样的模型是高度可解释的。我们通过证明该框架可以通过采用标准跳跃扩散形式的模型来优雅地表达，从而进一步确保数学一致性。此外，我们考虑了一种马尔可夫调制方法，用于在以随机参数为特征的状态之间进行切换。对于所有考虑的模型，我们推导特征函数，提供具有广泛应用的多功能工具。在数值实验中，我们将该框架应用于期权定价的金融问题。在双态模型中分析了参数不确定性的影响，其中资产过程在分别充满高不确定性和低不确定性的高波动性和低波动性时期之间切换。